Introduction

The purpose of this project is to study the relationship between life expectancy and other statistics of countries. Life expectancy is an estimate of how long a person would live on average for each country. As life expectancy is a very point indicate of people life quality, this study will help people understand how to improve people life quality. The data of this project comes from the Global Health Observatory (GHO) data repository under World Health Organization (WHO). Globally, life expectancy has increased by more than 6 years between 2000 and 2019 – from 66.8 years in 2000 to 73.4 years in 2019. And we hope to find what’s the most important issue behind that increase for life expectancy.

This project contains 2 parts:

The first part is to explore the relationships between these variables for model insight.

The second part is to build models to find the important predictors of life expectancy.

Data

Here we load in the data, and present the first several rows of the data

Life_Exp <- read_csv(file = "Life Expectancy Data.csv") %>% na.omit() 
Life_Exp %>% head()

## # A tibble: 6 x 22
##   Country  Year Status `Life expectan~` `Adult Mortali~` `infant deaths` Alcohol
##   <chr>   <dbl> <chr>             <dbl>            <dbl>           <dbl>   <dbl>
## 1 Afghan~  2015 Devel~             65                263              62    0.01
## 2 Afghan~  2014 Devel~             59.9              271              64    0.01
## 3 Afghan~  2013 Devel~             59.9              268              66    0.01
## 4 Afghan~  2012 Devel~             59.5              272              69    0.01
## 5 Afghan~  2011 Devel~             59.2              275              71    0.01
## 6 Afghan~  2010 Devel~             58.8              279              74    0.01
## # ... with 15 more variables: `percentage expenditure` <dbl>,
## #   `Hepatitis B` <dbl>, Measles <dbl>, BMI <dbl>, `under-five deaths` <dbl>,
## #   Polio <dbl>, `Total expenditure` <dbl>, Diphtheria <dbl>, `HIV/AIDS` <dbl>,
## #   GDP <dbl>, Population <dbl>, `thinness  1-19 years` <dbl>,
## #   `thinness 5-9 years` <dbl>, `Income composition of resources` <dbl>,
## #   Schooling <dbl>

The dataset contains 1649 observations and 22 variables. These 1649 observations are from 133 countries, and recorded their life expectancy from 2000 to 2015. Some countries miss some year.

The response variable of this dataset is Life expectany. This is a continuous variable ranged from 0 to 100.

The predictor variables have two main types:

Numeric variables:

- Years: ranged from 2000 to 2015
- Statistics about longetivity and mortality: Adult Mortality, infant deaths, under-five deaths, Population
- Statistics about economy: Total expenditure, GDP, Income composition of resources
- Statistics about illness: Hepatitis B, Measles, Polio, Diphtheria, HIV/AIDS
- Statistics about people: Schooling, thinness 1-19 years, thinness 5-9 years, BMI , Alcohol.

Category variable:

- Country
- Status: Developing, Developed

For the detailed description of these data, the Adult.Mortality column represents the adult mortality rates of both genders, which is the probability of dying between 15 and 60 years per 1000 population. Infant.deaths shows the number of infant deaths per 1000 population. The Alcohol column describes the total litres of consumption for pure alcohol recorded per capita for ages 15 and older. The Hepatitis.B, Polio, and Diphteria variables shows the percentage of immunization coverage among 1-year-olds for these diseases. The Measles column represents the number of reported cases of the measles per 1000 population. thinness 1-19 years represents the percentage of thinness present in children ranging from the age of 10 to 19 years old. thinness 5-9 years represents the percentage of thinness present in children ranging from the age of 10 to 19 years old.

Thelife expectancy ranges from 44 to 89, so the people in the country with the highest life expectancy live almost twice as long as the ones in the country with the lowest life expectancy.

Simple Analysis of Response Variable

Life_Exp %>%
  summarise(average=mean(`Life expectancy`), 
            maximum=max(`Life expectancy`),
            minimum=min(`Life expectancy`))

## # A tibble: 1 x 3
##   average maximum minimum
##     <dbl>   <dbl>   <dbl>
## 1    69.3      89      44

The life expectancy ranges from 44 to 89, so the people in the country with the highest life expectancy live almost twice as long as the ones in the country with the lowest life expectancy. Also, the average life expectancy is 69.3023, which is close to 70. This value is much higher than the past decades. Thus, it’s significant to explore these situations and predict the future possibility.

Explorarotory data analysis

Life Expectancy

First, we can use histogram to roughly analyze the response variable “Life Expectancy” in the dataset, and later we will predict the life expectancy in our model.

ggplot(Life_Exp, aes(`Life expectancy`)) +
  geom_histogram(bins = 50, color = "white") +
  labs(title = "Histogram of Life Expectancy") +
  theme_bw()

From the histogram above, we can see that the distribution of life expectancy is a little light skewed. The range of life expectancy is from 44 to 89, and most life expectancy are concentrated from 60 to 80. Especially from 70 to 75, there is the highest count. Besides, the life expectancy, which are more than 85 and less than 50, have less count. Next, we will continue to explore the relationship between the life expectancy and other factors, which may affect the life expectancy.

Development status

For the next question, Do developed countries significantly have higher life expectancy than the undeveloped countries? Regarding life expectancy and developed/developing countries, we can use EDA method and statistic method to answer this question and explore the relationship between the life expectancy and development status.

First, we can use the histogram to directly display the the number of developed/developing countries.

status_counting <- Life_Exp %>%
    group_by(Status) %>%
    summarise(Count = n())
ggplot(data = status_counting, aes(x = Status, y = Count)) +
  geom_histogram(bins = 50, stat = "identity", fill = c("red", "blue")) +
  labs(title="Histogram of Development Status") +
  theme_bw()

From the histogram we can see that the statistics of developing countries (more than 1250)is much more than developed countries (less than 250). The big difference is formed because there are more developing countries in the world.

Then we can use a group boxplot to further illustrate that the developed countries has higher life expectancy. Here is the Life Expectancy against Development Status group box plot.

library(ggstatsplot)
plt <- ggbetweenstats(
  data = Life_Exp,
  x = Status, 
  y = "Life expectancy",
  plot.type = "box",
  type = "p",
  conf.level = 0.99,
  title = "Parametric test",
  bf.message = FALSE,
  results.subtitle = FALSE
)
plt <- plt + 
  # Add labels and title
  labs(
    x = "Development Status",
    y = "Life Expectancy",
    title = "Life Expectancy against Development Status"
  )
plt

From the boxplot, we can see there is a stark difference in the life expectancy in developing and developed countries. The mean of developed countries is 78.69, and the mean of developing countries is 67.69. The median of developed countries is around 78, and the median of developing countries is around 70. Thus, Both the mean and median for developed countries life expectancy is much higher than the developing ones. Besides, the range of life expectancy for developed countries is around between 70 and 90. The range of life expectancy for developing countries is around between 45 and 90. Thus, The minimum life expectancy is also much lower in developing countries.

We can also apply parametric t test to illustrate that difference is significant.

print("result of t test for difference of 2 groups")

## [1] "result of t test for difference of 2 groups"

t.test(Life_Exp$`Life expectancy`[Life_Exp$Status == "Developing"],
       Life_Exp$`Life expectancy`[Life_Exp$Status == "Developed"],
       alternative = "less")

## 
##  Welch Two Sample t-test
## 
## data:  Life_Exp$`Life expectancy`[Life_Exp$Status == "Developing"] and Life_Exp$`Life expectancy`[Life_Exp$Status == "Developed"]
## t = -31.117, df = 616.28, p-value < 2.2e-16
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##       -Inf -10.42181
## sample estimates:
## mean of x mean of y 
##  67.68735  78.69174

As the p value for the t test is smaller than 0.05, so we can reject the null under 0.05 significance level and conclude that the true difference between developing countries and developed countries is less than 0. Thus, we can see that the status of the country could affect life expectancy greatly. The people in the developed countries have larger life expectancy than the people in the developing countries.

Country

Then, we will explore the relationship between the life expectancy and countries. Different from only analyzing the development status, this part we will analyze the life expectancy among each different country in the dataset.

We can calculate the mean life expectancy for all countries, and then plot that value on the global map.

# obtain the mean by country
new_data <- Life_Exp
summary_data <- new_data %>% 
    group_by(Country) %>% 
    summarize(mean=mean(`Life expectancy`))
library(plotly)
plot <- plot_geo(summary_data, locationmode="country names")%>%
    add_trace(locations=~Country,
             z=~mean,
             color=~mean) %>%
    plotly::layout(autosize = T,
                   title = 'Mean Life Expectancy from 2000-2015 in map',
                   geo = list(showframe = TRUE,
                              showcoastlines = TRUE,
                              projection = list(type = 'Mercator')))
plot

From this global map, we can see that African countries especially mid-south African countries are almost the lowest life expectancy regions, and the Canada, Europe and Australia are the highest regions.

Also, We will use bar plot to find the countries which has the highest and lowest life expectancy.

options(repr.plot.width = 20, repr.plot.height = 20) 
Life_Exp %>%
  group_by(Country) %>%
  summarise(mean = mean(`Life expectancy`)) %>%
  ggplot(aes(x = mean, y = reorder(Country, mean))) +
  geom_bar(stat = 'identity') +
  labs(title = 'Mean Life Expectancy from 2000-2015', 
       x = 'Life Expectancy',
       y = 'Countries')

summary_data %>%
  arrange(desc(mean))

## # A tibble: 133 x 2
##    Country      mean
##    <chr>       <dbl>
##  1 Ireland      83.4
##  2 Canada       82.2
##  3 France       82.2
##  4 Italy        82.2
##  5 Spain        82.0
##  6 Australia    81.9
##  7 Sweden       81.9
##  8 Austria      81.5
##  9 Netherlands  81.3
## 10 Greece       81.2
## # ... with 123 more rows

From the bar plot above, we can see that Ireland has the highest Life Expectancy and Sierra Leone has the lowest Life Expectancy. This result of bar graph is consistent withe the calculation result. Also, Ireland is a European country, and Sierra Leone is a African country. Thus, this result is also consistent with the analysis of global map above.

Selected num variables analysis

Now let’s make a scatter plot matrix of the response variable life expectancy with some selected continuous variable: Year, GDP, Population, Schooling, thinness 1-19 years.

library(GGally)
ggpairs(Life_Exp %>%
          select(`Life expectancy`, Year, GDP, 
                 Population, Schooling, `thinness  1-19 years`))

From the scatter plot matrix, we can observe there is a slightly positive relationship between year and life expectancy.The life expectancy increase when year increase. Lowest life expectancy increase much more when GDP increase, so the GDP has a high effect on the life expectation. The life expectancy is not so affected by population. Very suprisingly, the schooling has the most positive correlation with life expectancy as 0.752, and from the scatter plot matrix there is a very clear linear trend between them. For the thinness 1-19 years, it is not a suprise to observe it has a slightly negative correlation with life expectancy.

Overview all num variables with life expectancy

Then, we will independently create the scatterplot for each predictor to clearly and directly determine the relationship between life expectancy and numeric variables. Also, the overview will help the final summary from EDA better.

Life_Exp %>%
  ggplot(aes(x=Year, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Year vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a slightly positive relationship between Year and Life Expectancy. As Year increase, the life expectancy will slightly increase.

Life_Exp %>%
  ggplot(aes(x=`Adult Mortality`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Adult Mortality vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a obviously negative relationship between Adult Mortality and Life Expectancy. As adult mortality increase, the life expectancy will decrease.

Life_Exp %>%
  ggplot(aes(x=`infant deaths`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Infant Deaths vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a slightly negative relationship between Infant Deaths and Life Expectancy. As infant deaths increase, the life expectancy will decrease.

Life_Exp %>%
  ggplot(aes(x= Alcohol, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Alcohol vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a positive relationship between Alcohol and Life Expectancy. As Alcohol increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x=`percentage expenditure`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Percentage Expenditure vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a positive relationship between percentage expenditure and Life Expectancy. As percentage expenditure increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x=`Hepatitis B`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Hepatitis B vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a slightly positive relationship between Hepatitis B and Life Expectancy. As Hepatitis B increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x= Measles, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Measles vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a slightly negative relationship between Measles and Life Expectancy. As Measles increase, the life expectancy will decrease.

Life_Exp %>%
  ggplot(aes(x= BMI, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of BMI vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a obviously positive relationship between BMI and Life Expectancy. As BMI increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x= `under-five deaths`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of under-five deaths vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a slightly negative relationship between under-five deaths and Life Expectancy. As under-five deaths increase, the life expectancy will decrease.

Life_Exp %>%
  ggplot(aes(x= Polio, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Polio vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a slightly positive relationship between Polio and Life Expectancy. As Polio increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x= `Total expenditure`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Total expenditure vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a slightly positive relationship between Total expenditure and Life Expectancy. As Total expenditure increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x= Diphtheria, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Diphtheria vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a slightly positive relationship between Diphtheria and Life Expectancy. As Diphtheria increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x=`HIV/AIDS`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of HIV/AIDS vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a obviously negative relationship between HIV/AIDS and Life Expectancy. As HIV/AIDS increase, the life expectancy will decrease.

Life_Exp %>%
  ggplot(aes(x= GDP, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of GDP vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a positive relationship between GDP and Life Expectancy. As GDP increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x= Population, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Population vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is almost no relationship between Population and Life Expectancy.

Life_Exp %>%
  ggplot(aes(x=`thinness  1-19 years`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of thinness  1-19 years vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a obviously negative relationship between thinness 1-19 years and Life Expectancy. As thinness 1-19 years increase, the life expectancy will decrease.

Life_Exp %>%
  ggplot(aes(x=`thinness 5-9 years`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of thinness 5-9 years vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a obviously negative relationship between thinness 5-9 years and Life Expectancy. As thinness 5-9 years increase, the life expectancy will decrease.

Life_Exp %>%
  ggplot(aes(x= `Income composition of resources`, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Income composition of resources vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a obviously positive relationship between Income composition of resources and Life Expectancy. As Income composition of resources increase, the life expectancy will increase.

Life_Exp %>%
  ggplot(aes(x= Schooling, y=`Life expectancy`)) + 
  geom_point(alpha = 0.2) + 
  labs(title = 'scatterplot of Schooling vs Life Expectancy') +
  geom_smooth(method = 'lm', formula = 'y ~ x') +
  theme_bw() +
  theme(plot.title = element_text(hjust = 0.5))

There is a obviously positive relationship between Schooling and Life Expectancy. As Schooling increase, the life expectancy will increase.

Correlation of all num variables

First, we obtain the correlation matrix of all the continuous variables except time to take a rough look at positive or negative correlation.

library(corrr)
cor_life_exp <- Life_Exp %>%
  select(-Country, -Year, -Status) %>%
  correlate()
rplot(cor_life_exp)

Next, we create a visualization of the matrix to further analyze the correlation among the variables. (With the specific correlation number, we can determine their correlation specifically.)

ggcorr(
  Life_Exp %>% select(-Country, -Year, -Status),
  cor_matrix = cor(Life_Exp %>% select(-Country, -Year, -Status), use = "pairwise"),
    label = T, 
    hjust = 0.95,
    angle = 0,
    size = 4,
    layout.exp = 3
)

There are several things to state:

For the response variable life expectancy, the most negative correlated predictors are adult mortality, Hiv-aids and thinness. The most positive correlated predictors are schooling, income composition of resources, BMI and GDP.
The infant death has 1 correlation with under five deaths, so we should only include the under 5 death. Also, for the 2 thinness variables, we should only include 1 as their correlation is 0.9. Same with the percentage expenditure and GDP. We should not include all of these predictors, because otherwise colinearlity will be introduced.

Then when we exclude these variables, we can obtain the remaining variables correlation network plot.

corrr::correlate(Life_Exp %>%
   select(-Country, -Year, -Status, -`infant deaths`,
   - `thinness 5-9 years`, -`Total expenditure`)) %>% corrr::network_plot()

From this network we can observe that the 3 vaccinate variable: Polio, Diphtheria and Hepatitis B are close in correlation. GDP, percentage expenditure, schooling, Alcohol, BMI are close in correlation. Measles, under 5 deaths and thiness are close in correlation.

Summary from EDA

The variables that life expectancy is most positively related are: schooling, and income composition of resources, BMI. The BMI seems a little bit wierd, because we know higher BMI will indicate obesity. This may correlated with other economy indicators.
The variables most negatively associated with life expectancy are: adult mortality and HIV/AIDS, thinness. These are not a surprise.
Life expectancy does not have much correlation with population.
Counties in Europe, Oceania, and North America has higher life expectancy, and countries in Africa has lower life expectancy.
Developed countries has higher life expectancy than undeveloped countries.
Life Expectancy range from 44 to 89, and most concentrated on from 70 to 75.

Preparation for modeling

Preparing the data

Life_Exp  <- Life_Exp  %>%
  mutate(Country = factor(Country)) %>%
  mutate(Status = factor(Status)) 
head(Life_Exp)

## # A tibble: 6 x 22
##   Country  Year Status `Life expectan~` `Adult Mortali~` `infant deaths` Alcohol
##   <fct>   <dbl> <fct>             <dbl>            <dbl>           <dbl>   <dbl>
## 1 Afghan~  2015 Devel~             65                263              62    0.01
## 2 Afghan~  2014 Devel~             59.9              271              64    0.01
## 3 Afghan~  2013 Devel~             59.9              268              66    0.01
## 4 Afghan~  2012 Devel~             59.5              272              69    0.01
## 5 Afghan~  2011 Devel~             59.2              275              71    0.01
## 6 Afghan~  2010 Devel~             58.8              279              74    0.01
## # ... with 15 more variables: `percentage expenditure` <dbl>,
## #   `Hepatitis B` <dbl>, Measles <dbl>, BMI <dbl>, `under-five deaths` <dbl>,
## #   Polio <dbl>, `Total expenditure` <dbl>, Diphtheria <dbl>, `HIV/AIDS` <dbl>,
## #   GDP <dbl>, Population <dbl>, `thinness  1-19 years` <dbl>,
## #   `thinness 5-9 years` <dbl>, `Income composition of resources` <dbl>,
## #   Schooling <dbl>

Splitting the data

set.seed(123)
ec_split <- Life_Exp %>%
  initial_split(prop = 0.8, strata = "Status")
ec_train <- training(ec_split)
dim(ec_train)

## [1] 1318   22

ec_test <- testing(ec_split)
dim(ec_test)

## [1] 331  22

Here split the data with a 0.8 proportion, strata by Status. Training set include 1318 observations and testing include 331 observations

Making the recipe and folds

ec_recipe <- recipe(`Life expectancy` ~ `Status` + `Adult Mortality` + `infant deaths` +
                      Alcohol + `percentage expenditure` + `Hepatitis B` + Measles + BMI +
                      `under-five deaths` + Polio + `Total expenditure`+
                      Diphtheria+ `HIV/AIDS` + GDP + Population + `thinness  1-19 years` + 
                      `thinness 5-9 years` + `Income composition of resources` + Schooling,
                    data = ec_train) %>%
  step_dummy(all_nominal_predictors()) %>%
  step_normalize(all_predictors()) %>%
  step_novel(all_nominal_predictors()) %>%
  step_zv(all_nominal_predictors())
               
ec_folds <- vfold_cv(ec_train, strata = Status, v = 10, repeats = 5)

Ridge Regression

Here we use the Ridge Regression Model to fit the life expectancy. Ridge regression is one of the main types of the Regularization approach to select features in regression. Ridge regression minimizes the sum of squared residuals and \(\lambda * ||\beta||^2\).

ridge_spec <- linear_reg(penalty = tune(), mixture = 0) %>%
  set_mode("regression") %>%
  set_engine("glmnet")
ridge_workflow <- workflow() %>%
  add_recipe(ec_recipe) %>%
  add_model(ridge_spec)
set.seed(123)
penalty_grid <- grid_regular(penalty(range = c(-4, 4), trans = log10_trans()), levels = 20)
tune_res <- tune_grid(
  ridge_workflow,
  resamples = ec_folds,
  grid = penalty_grid
)
autoplot(tune_res)

Now calculate the metrics of our regression tune and look at the mean and standard error.

Ridge_RMSE <- collect_metrics(tune_res) %>%
  dplyr::select(.metric, mean, std_err)
Ridge_RMSE = Ridge_RMSE[c(1,2),]

Now select the best ridge model based on R square metric and fit the model, and get the final prediction result.

best_penalty <- select_best(tune_res, metric = "rsq")
best_penalty

## # A tibble: 1 x 2
##   penalty .config              
##     <dbl> <chr>                
## 1  0.0001 Preprocessor1_Model01

ridge_final <- finalize_workflow(ridge_workflow, best_penalty)
ridge_final_fit <- fit(ridge_final, data = ec_train)
Ridge_Prediction <- predict(ridge_final_fit, new_data = ec_test %>% dplyr::select(-`Life expectancy`))
Ridge_Prediction <- bind_cols(Ridge_Prediction, ec_test %>% dplyr::select(`Life expectancy`))
Ridge_Graph <- Ridge_Prediction %>%
  ggplot(aes(x=.pred, y=`Life expectancy`)) + geom_point(alpha = 1) + geom_abline(lty = 2) + theme_bw() + coord_obs_pred()
Ridge_Accuracy <- augment(ridge_final_fit, new_data = ec_test) %>%
  rsq(truth = `Life expectancy`, estimate = .pred)

Lasso Regression

The second model is a Lasso Regression Model. Lasso minimizes the sum of squared residuals and \(\lambda * ||\beta||\).

lasso_spec <-
  linear_reg(penalty = tune(), mixture = 1) %>%
  set_mode("regression") %>%
  set_engine("glmnet")
lasso_workflow <- workflow() %>%
  add_recipe(ec_recipe) %>%
  add_model(lasso_spec)
set.seed(1234)
tune_res_lasso <- tune_grid(
  lasso_workflow,
  resamples = ec_folds,
  grid = penalty_grid
)
autoplot(tune_res_lasso)

Lasso_RMSE <- collect_metrics(tune_res_lasso) %>%
  dplyr::select(.metric, mean, std_err) %>%
  head(2)

We collect the metrics of our Lasso regression tune and look at the mean and standard error. Now select the best ridge model based on R square metric and fit the model, and get the final prediction result.

best_penalty_lasso <- select_best(tune_res_lasso, metric = "rsq")
lasso_final <- finalize_workflow(lasso_workflow, best_penalty_lasso)
lasso_final_fit <- fit(lasso_final, data = ec_train)
Lasso_Prediction <- predict(lasso_final_fit, new_data = ec_test %>% dplyr::select(-`Life expectancy`))
Lasso_Prediction <- bind_cols(Lasso_Prediction, ec_test %>% dplyr::select(`Life expectancy`))
Lasso_Graph <- Lasso_Prediction %>%
  ggplot(aes(x=.pred, y=`Life expectancy`)) + geom_point(alpha=1) + geom_abline(lty = 2) + theme_bw() + coord_obs_pred()
Lasso_Accuracy <- augment(lasso_final_fit, new_data = ec_test) %>%
  rsq(truth = `Life expectancy`, estimate = .pred)

Boosted Model

The third model is a boosted tree model. A boosted model builds a weak decision tree that has low predictive accuracy. Then the method sequentially improving previous decision trees.

boost_spec <- boost_tree() %>%
  set_engine("xgboost") %>%
  set_mode("regression")
boost_wf <- workflow() %>%
  add_model(boost_spec %>%
  set_args(trees = tune())) %>%
  add_recipe(ec_recipe)
set.seed(123)
boost_grid <- grid_regular(trees(range = c(10, 2000)), levels = 50)
boost_tune_res <- tune_grid(
  boost_wf,
  resamples = ec_folds,
  grid = boost_grid,
)
autoplot(boost_tune_res)

Boost_RMSE <- collect_metrics(boost_tune_res) %>% 
  dplyr::select(.metric, mean, std_err) %>%
  head()

We collect the metrics of our regression tune and look at the mean and standard error. Now select the best ridge model based on R square metric and fit the model, and get the final prediction result.

best_boost_final <- select_best(boost_tune_res,metric = "rsq")
best_boost_final_model <- finalize_workflow(boost_wf, best_boost_final)
best_boost_final_model_fit <- fit(best_boost_final_model, data = ec_train)
Boost_Prediction <- predict(best_boost_final_model_fit, new_data = ec_test %>% dplyr::select(-`Life expectancy`))
Boost_Prediction <- bind_cols(Boost_Prediction, ec_test %>% dplyr::select(`Life expectancy`))
Boost_Graph <- Boost_Prediction %>%
  ggplot(aes(x=.pred, y=`Life expectancy`)) + geom_point(alpha=1) + geom_abline(lty = 2) + theme_bw() + coord_obs_pred()
Boost_Accuracy <- augment(best_boost_final_model_fit, new_data = ec_test) %>%
  rsq(truth = `Life expectancy`, estimate = .pred)

Decision - Tree model

The fourth model is a decision tree model.

tree_spec <-decision_tree() %>%
  set_engine("rpart")
class_tree_spec <- tree_spec %>%
  set_mode("regression")
  
class_tree_wf <- workflow() %>%
  add_model(class_tree_spec %>% set_args(cost_complexity = tune())) %>%
  add_recipe(ec_recipe)
set.seed(123)
param_grid <- grid_regular(cost_complexity(range = c(-3, 3)), levels = 10)
tune_res_tree <- tune_grid(
  class_tree_wf,
  resamples = ec_folds,
  grid = param_grid,
)
autoplot(tune_res_tree)

Tree_RMSE <- collect_metrics(tune_res_tree) %>%
  dplyr::select(.metric, mean, std_err) %>%
  head(2)

We collect the metrics of our regression tune and look at the mean and standard error. Now select the best ridge model based on R square metric and fit the model, and get the final prediction result.

library(rpart.plot)
best_complexity <- select_best(tune_res_tree, "rsq")
class_tree_final <- finalize_workflow(class_tree_wf, best_complexity)
class_tree_final_fit <- fit(class_tree_final, data = ec_train)
class_tree_final_fit %>%
  extract_fit_engine() %>%
  rpart.plot()

From the plot of the decision tree, the most important variables are: income composition of resources, HIV/AIDs. Now we predict the life expectancy on the test set.

Tree_Prediction <- predict(class_tree_final_fit, new_data = ec_test %>% dplyr::select(-`Life expectancy`))
Tree_Prediction <- bind_cols(Tree_Prediction, ec_test %>% dplyr::select(`Life expectancy`))
Tree_Graph <- Tree_Prediction %>%
  ggplot(aes(x=.pred, y=`Life expectancy`)) + geom_point(alpha=1) + geom_abline(lty = 2) + theme_bw() + coord_obs_pred()
Tree_Accuracy <- augment(class_tree_final_fit, new_data = ec_test) %>%
  rsq(truth = `Life expectancy`, estimate = .pred)

Model comparison

Now we compare the four different models: ridge, lasso, boost and decision tree. We will compare the four different models by these factors: - Prediction Graphs - RMSE & RSQ (R-Squared) from Training Set - RSQ from Testing Set

Prediction Graphs

library(ggpubr)
figure <- ggarrange(Ridge_Graph, Lasso_Graph, Boost_Graph,Tree_Graph,
                    labels = c("Ridge", "Lasso", "Boost","Tree"),
                    ncol = 2, nrow = 2)
figure

In the plots the dotted line represents where the points would be if the model prediction is the same as true life expectancy. Looking at the plots it seems that the Boost model has the points closest to the dotted line, which means it has the best performance in the four models.

RMSE & RSQ in Training Set

Ridge regression model:

head(Ridge_RMSE)

## # A tibble: 2 x 3
##   .metric  mean std_err
##   <chr>   <dbl>   <dbl>
## 1 rmse    3.72  0.0391 
## 2 rsq     0.826 0.00415

Lasso regression model:

head(Lasso_RMSE)

## # A tibble: 2 x 3
##   .metric  mean std_err
##   <chr>   <dbl>   <dbl>
## 1 rmse    3.64  0.0371 
## 2 rsq     0.834 0.00419

Boost model:

head(Boost_RMSE, 2)

## # A tibble: 2 x 3
##   .metric  mean std_err
##   <chr>   <dbl>   <dbl>
## 1 rmse    3.10  0.0403 
## 2 rsq     0.932 0.00258

Decision Tree:

head(Tree_RMSE,2 )

## # A tibble: 2 x 3
##   .metric  mean std_err
##   <chr>   <dbl>   <dbl>
## 1 rmse    2.78  0.0434 
## 2 rsq     0.902 0.00346

So the Boost model has the highest RSQ on the training set, decision tree has the smallest rmse on the training set. Based on the RSQ, boost model is the best.

R-Squared of Testing Set

rsq_comparisons <- bind_rows(Ridge_Accuracy, Lasso_Accuracy, Boost_Accuracy, Tree_Accuracy) %>% 
  tibble() %>% mutate(model = c("Ridge", "Lasso", "Boost", "Tree")) %>% 
  dplyr::select(model, .estimate) %>%
  arrange(.estimate)
rsq_comparisons

## # A tibble: 4 x 2
##   model .estimate
##   <chr>     <dbl>
## 1 Ridge     0.816
## 2 Lasso     0.821
## 3 Tree      0.902
## 4 Boost     0.949

Looking at the R-Squared of the test set. Of the 4 models, the Boost model has the highest R squared 0.948, much higher than the others.

Conclusion

From our exploratory data analysis and model fitting part, we can know that the Boost model performs much better than the other methods. We also checked that developing countries have on average lower life expectancy than developed countries.

The next steps can be the longitudinal data analysis. Because this data has a covariate time, and this data is a longitudinal data, so some complex model such as linear mixed model (random effects model).

Final project

Yu Tian

6/10/2022